The circle problem of gauss and the divisor problem of dirichlet—still unsolved

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Abstract

Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for Σnx r2(n) and Σnx d(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially “know” what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan’s contributions.

Original languageEnglish (US)
Pages (from-to)99-114
Number of pages16
JournalAmerican Mathematical Monthly
Volume125
Issue number2
DOIs
StatePublished - 2018

ASJC Scopus subject areas

  • Mathematics(all)

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